A very interesting question. It depends entirely on who you are, who your congregation is, and the purpose of using Gregorian Chant for the office. From my perspective, as both an Anglican and a plainsong 'purist', there is no single 'correct' way to sing Mattins or Evensong to Gregorian chant.

The Hymnal 1940 provides plainsong settings for the Venite (#611), Te Deum (#621), and Benedictus (#641), as well as for the Magnificat (#658) and Nunc dimittis (#673). As you mentioned, rubrics for both Venite and Te Deum say that the intonation should be sung only for the first verse - but in my opinion this is because the canticles are being treated as psalms, where they normally would have their own proper chants. The settings of the Benedictus, Magnificat, and Nunc dimittis, on the other hand, include the intonation at the beginning of each verse, as is the normal custom for Gospel canticles.

If I were to present the Te Deum in a psalm-tone setting, I would prefer to use the intonation for each verse, on a par with the Benedictus - I think the dignity of the canticle demands it. For the Venite, I would say whatever makes the most sense for you.